Sttne 

Standardized  Reasoning  Tests 
in  Arithmetic  and  H»w 
t*  Utilize  Them 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


'A    'N 


STANDARDIZED    REASONING   TESTS 

IN  ARITHMETIC  AND   HOW 

TO  UTILIZE  THEM 


BY  CLIFF  W.  STONE,  PH.D. 

PROFESSOR  OF  EDUCATION,  STATE  COLLEGE  OF  WASHINGTON 

PULLMAN,    WASHINGTON 

FORMERLY  DIRECTOR  OF  TEACHING,  IOWA  STATE  TEACHERS  COLLEGE 
CEDAR    FALLS,    IOWA 


T*"'*  "  T  the  last  date  ' 

TEACHERS   COLLEGE,   COLUMBIA   UNIVERSITY 

CONTRIBUTIONS  TO  EDUCATION,   NO.   83 
SECOND  EDITION,  REVISED  AND  ENLARGED 


SOUTHERN  BRANCH, 

UNIVERSITY  Of-  CA!  TOfiNIA, 

LIBRA-/, 
iLOS  ANt^L^c,  CAL:,-. 


44905 


PUBLISHED  BY 

Irarhpra  CulUgr,  Calnmbta  Xnittfrattg 

NEW  YORK  CITY 
1921 


Copyright,  1916  and  1921,  by  CLIFF  W.  STONE 


^35 


TABLE  OF  CONTENTS 

PAGE 

INTRODUCTION  v 

I.    THE  TESTS.  .  1 


Test  in  "Fundamentals" 
Reasoning  Test  I 
Reasoning  Test  II 
Preliminary  Test 

II.    CONDITIONS  AND  DIRECTIONS  FOR  USING  THE  TEST? 

Points  to  be  Observed 
How  to  Give  the  Tests 
Directions  to  Pupils 

III.    DIRECTIONS  FOR  SCORING  THE  TESTS  . . 


On  Basis  of  Answers 

On  Basis  of  Reasoning — Not  Counting  Partial  Solutions 

On  Basis  of  Reasoning — Counting  Partial  Solutions 

Scoring  for  Accuracy 

Aids  to  Scoring  Test  I 

Aids  to  Scoring  Test  II 

IV.    STANDARDS 14 

Sources  of  Data  for  Standards 

Large  Number  of  Pupils 

Results  of  Diagnostic  Measuring  and  Remedial  Teaching 
Standards  Adopted 

For  Median  Scores  on  Basis  of  Answers — Table  I 

For  Median  Scores  on  Basis  of  Reasoning — Table  II 

For  Accuracy  of  Reasoning-— Table  III 

V.    REPRESENTING  SCORES 17 

Importance  and  Means  of  Adequate  Representation 
A  Simple  Tabular  Representation — The  Springfield  Survey  Plan 
Representing  Scores  as  Aids  in  Class  Diagnosis  and  General  Su- 
pervision 

Representations  of  Percentages  of  Pupils  Making  Given  Scores 
The  Butte  Plan 
Graph  Form 

Procedure  in  Filling  Form 
Graph  of  VI  A,  Bloomington,  Indiana,  Scores 
Percentage  Graph  of  Iowa  State  Teachers  College  Scores 
The  Bloomington,  Indiana,  Progress  Graph 
Record  of  Hackensack,  New  Jersey,  Improvement 
Representing  Scores  as  Aids  in  Individual  Diagnosis  and  Remedial 

Teaching 

Scores  of  Individual  Pupils 
Accuracy  of  Individual  Pupils 
Both  Scores  and  Accuracy  of  Individual  Pupils 


iv  Table  of  Contents 

VI.    UTILIZING  RESULTS 27 

Purposes  of  Measuring 
Utilizing  Results  of  Measuring 

Teacher's  Use  Compared  with  That  of  Physician 

Sample  Solutions  of  Stone  Reasoning  Problems 

Class  Diagnosis  Record  Sheet 

Individual  Diagnosis  Record  Sheet 
Two  Dangers 
Possible  Causes  of  Failure 
Securing  Improvement 


INTRODUCTION 

The  occasion  for  publishing  this  manual  of  directions  for  the 
Stone  Reasoning  Tests  is  the  exhaustion  of  the  edition  of  the 
book,  Arithmetical  Abilities  and  Some  Factors  Determining 
Them,1  in  which  Reasoning  Test  I  first  appeared.  Then,  too,  it 
has  been  found  that  the  brief  record  of  the  test  as  given  in  the 
original  book  does  not  give  adequate  help  for  those  who  may  wish 
to  measure  the  reasoning  ability  of  children.  It  is  hoped  that 
the  publication  of  this  manual  will  furnish  additional  help  in 
supervising  and  in  teaching  the  reasoning  phases  of  arithmetic. 

Another  purpose  of  publishing  this  manual  is  that  of  further- 
ing, in  some  measure,  progress  in  the  scientific  study  of  the 
teaching  of  arithmetic.  This  progress  will  doubtless  move  along 
two  lines,  viz.,  the  development  and  use  of  better  and  more 
complete  tests,  and  the  settling  of  some  of  the  many  problematic 
conflicts  in  the  teaching  of  arithmetic.2 


1  Stone,  Cliff  W.,  Arithmetical  Abilities  and  Some  Factors  Determining 
Them.    This  book  is  out  of  print  but  may  be  borrowed  from  libraries  oi 
teachers   colleges   or   schools    of    education.     Extended    extracts    may   be 
found   in   Strayer's   Teaching  Process;   Thorndike   and    Strayer's   Educa- 
tional Administration;  Brown  and  Coffman's  Plow  to  Teach  Arithmetic; 
and  Howell's  A  Fundamental  Study  in  the  Teaching  of  Arithmetic. 

2  Some  of  these  conflicts  are  stated  in  an  article  by  the  author  entitled, 
Problems  in  Scientific  Study  of  the  Teaching  of  Arithmetic,  Journal  of 
Educ.  Psychology,  Vol.  IV,  No.  i,  pp.  1-16. 


SECTION  I 
THE  TESTS 

As  originally  developed  and  included  in  Arithmetical  Abilities 
and  Some  Factors  Determining  Them,  the  tests  were  two  in 
number,  a  test  in  reasoning  and  one  in  so-called  fundamentals. 
The  original  reasoning  test  has  proved  so  serviceable  that  a 
second  reasoning  test  has  been  developed.  It  is  known  as 
Reasoning  Test  II  and  with  proper  precautions  may  be  used 
as  an  equivalent  of  the  original  test,  Test  I. 

The  test  in  fundamentals  has  been  so  modified  and  improved 
by  Courtis  that  it  has  been  replaced  by  his  Series  B.1  However, 
for  the  benefit  of  those  who  may  wish  to  use  it,  the  author's 
original  test  in  fundamentals  is  here  reproduced  and  permission 
is  given  for  it  to  be  printed  so  that  each  pupil  may  have  a  copy. 
Reasoning  Tests  I  and  II  follow  on  pages  3  and  4.2 

Reasoning  Test  II  has  proved  somewhat  more  difficult  than 
the  original,  but  through  the  use  of  the  correction  formula 
printed  at  the  bottom  of  the  test  slip,  it  is  practically  an  equiva- 
lent. In  using  Test  II,  it  should  be  kept  in  mind  that  the  degree 
of  equivalence  is  much  closer  when  used  with  a  class  or  system 
of  50  or  more  pupils  than  when  used  with  a  less  number.  Warn- 
ing is  here  given  against  expecting  too  much  from  the  test  as 
an  exact  measure  of  the  progress  of  individual  pupils. 

The  accuracy  of  the  first  measurement  may  be  considerably 
increased  by  the  use  of  the  preliminary  test  shown  on  page  5. 
Through  such  a  test,  there  is  a  reduction  in  certain  gross  sources 
of  error,  such  as  strangeness,  misunderstanding,  nervousness,  etc. 

The  same  directions,  precautions,  etc.,  should  be  followed  in 
using  the  preliminary  test  as  in  using  the  regular  tests,  except  that 
the  pupils  may  be  told  that  it  is  preliminary. 


1  These  tests  may  be  secured  from  the  author,  S.  A.  Courtis,  241  Eliot 
St.,  Detroit,  Michigan.     They  are  also  included  in  the  Courtis  Standard 
Practice  Tests,  published  by  the  World  Book  Co.,  Yonkers,  N.  Y. 

2  Reasoning  Tests  I  and  II  may  be  obtained  from  the  Bureau  of  Pub- 
lications, Teachers  College,  525  West  120th  Street,  New  York  City. 

I 


2  Standardised  Reasoning  Tests 

TEST  IN  FUNDAMENTALS 

Work  as  many  of  these  problems  as  you  have  time  for;  work 
them  in  order  as  numbered. 

i.  Add  2375 

4052 

6354 
260 

5041 
1543 


2.  Multiply  3265  by  20. 

3.  Divide  3328  by  64. 

4.  Add  596 

428 
94 

75 
302 

645 
984 
897 


5.  Multiply  768  by  604. 

6.  Divide   1918962  by  543. 

7.  Add  4695 

872 
7948 
6786 

567 
858 

9447 
7499 


8.  Multiply  976  by  87. 

9.  Divide  2782542  by  679. 
10.  Multiply  5489  by  9876. 
IT.  Divide  5099941  by  749. 

12.  Multiply  876  by  79. 

13.  Divide  62693256  by  859. 

14.  Multiply  96879  by  896. 


The  Tests  3 

REASONING  TEST  I 

Solve  as  many  of  the  following  problems  as  you  have  time  for; 
work  them  in  order  as  numbered. 

1.  If  you  buy  2  tablets  at  7  cents  each  and  a  book  for  65  cents, 
how  much  change  should  you  receive  from  a  two-dollar  bill? 

2.  John  sold  4  Saturday  Evening  Posts  at  5  cents  each.     He 
kept  YZ  the  money  and  with  the  other  ^  he  bought  Sunday 
papers  at  2  cents  each.     How  many  did  he  buy? 

3.  If  James  had  4  times  as  much  money  as  George,  he  would 
have  $16.     How  much  money  has  George? 

4.  How  many  pencils  can  you  buy  for  50  cents  at  the  rate  of 
2  for  5  cents? 

5.  The  uniforms  for  a  baseball  nine  cost  $2.50  each.     The 
shoes  cost  $2  a  pair.     What  was  the  total  cost  of  uniforms  and 
shoes  for  the  nine? 

6.  In  the  schools  of  a  certain  city  there  are  2,200  pupils; 
YZ  are  in  the  primary  grades,  %  in  the  grammar  grades,  l/%  in 
the  high  school,  and  the  rest  in  the  night  school.     How  many 
pupils  are  there  in  the  night  school? 

7.  If  3^2  tons  of  coal  cost  $21,  what  will  5^  tons  cost? 

8.  A  newsdealer  bought  some  magazines  for  $i.     He  sold 
them  for  $1.20,  gaining  5  cents  on  each  magazine.     How  many 
magazines  were  there? 

9.  A  girl  spent  l/%  of  her  money  for  carfare,  and  three  times 
as  much  for  clothes.     Half  of  what  she  had  left  was  80  cents. 
How  much  money  did  she  have  at  first? 

10.  Two  girls  receive  $2.10  for  making  button-holes.     One 
makes  42,  the  other  28.     How  shall  they  divide  the  money? 

11.  Mr.  Brown  paid  y$  of  the  cost  of  a  building;  Mr.  John- 
son paid  l/2  the  cost'.     Mr.  Johnson  received  $500  more  annual 
lent  than  Mr.  Brown.     How  much  did  each  receive? 

12.  A  freight  train  left  Albany  for  New  York  at  6  o'clock. 
An  express  left  on  the  same  track  at  8  o'clock.     It  went  at  the 
rate  of  40  miles  an  hour.     At  what  time  of  day  will  it  overtake 
the  freight  train  if  the  freight  train  stops  after  it  has  gone  56 
miles  ? 


4  Standardised  Reasoning  Tests 

REASONING  TEST  II 

Solve  as  many  of  the  following  problems  as  you  have  time  for; 
work  them  in  order  as  numbered. 

1.  A  man  starting  on  a  journey  took  $200.     He  paid   for 
railroad  fare  $67 ;  for  berth  in  sleeping  car,  4  days,  $2  a  day ; 
hotel  bills,  15  days,  $3  a  day;  other  expenses,  $25.     How  much 
money  had  he  left? 

2.  Sam  had  12  marbles.     He  found  3  more  and  then  gave 
6  to  George.     How  many  did  Sam  have  left? 

3.  A  man  bought  163  barrels  of  flour  at  $11  a  barrel.    Fifteen 
barrels  were  spoiled  and  the  remainder  sold  at  $13  a  barrel.    Did 
he  gain  or  lose  and  how  much? 

4.  A  drover  bought   132  head  of  cattle  at  $45  a  head  and 
67  at  $61  a  head.    He  sold  them  at  $50  a  head.    Did  he  gain  or 
lose  and  how  much? 

5.  A  boy  had  $500  when  he  was  16  years  old  and  he  saved 
$100  a  year  until  he  was  20.     How  much  had  he  then? 

6.  If  a  bookkeeper  receives  $1400  a  year  for  his  services  and 
his  expenses  are  $840,  in  what  time  can  he  save  enough  to  buy 
42  acres  of  land  at  $140  an  acre? 

7.  Mr.  Brown  earns  $32  while  Mr.  Smith  earns  $21.     How 
much  money  will  Mr.  Brown  earn  while  Mr.  Smith  earns  $189? 

8.  A  school  in  a  certain  city  used  2516  pieces  of  chalk  in 
37  days.     Three  new  rooms  were  opened,  each  holding  50  chil- 
dren, and  the  school  was  then  found  to  use  87  pieces  of  chalk 
per  day.     How  many  more  sticks  of  chalk  were  used  per  day 
than  before? 

9.  Divide  $432  between  A  and  B  so  that  A  will  have  $16 
more  than  B. 

10.  Two  carpenters  receive  $154  for  the  work  they  do  on 
a  house.    One  worked  18  days  and  the  other  26  days.    How  much 
did  each  receive? 

11.  Mr.  Smith  owns  a  y\  interest  in  a  taxi  line  and  Mr.  Jones 
owns  the  rest.     Mr.  Jones  receives  $980  more  a  year  than  does 
Mr.  Smith.     How  much  does  each  receive?- 

12.  A  man  leaves  a  certain  place  at  8:00  A.  M.  and  travels 
due  east  at  the  rate  of  20  miles  per  hour.     One  hour  later  another 
man  follows  in  the  same  direction  at  the  rate  of  30  miles  per  hour. 
What  time  will  it  be  when  the  second  man  catches  the  first? 

NOTE:  With  the  time  limit,  all  other  conditions,  and  scoring  the  same  as 
for  Test  I,  this  one  has  proved  somewhat  more  difficult.  To  make  scores 
from  this  test  comparable  with  those  from  Test  I,  increase  scores  from 
this  test  as  follows  for  the  respective  grades : 

For  grade  V,  add  .5;  grade  VI,  .85;  grade  VII,  i.oo;  grade  VIII,  1.67. 


The  Tests  5 

A  PRELIMINARY  TEST* 

Solve  as  many  of  the  following  problems  as  you  have  time 
for;  work  them  as  numbered. 

1.  There  were  37  pupils  in  a  certain  sixth  grade;  22  were 
promoted  into  the  seventh  grade,  and   17  were  promoted  into 
the  sixth  grade  from  the  fifth  grade.     How  many  were  there  in 
the  sixth  grade? 

2.  A  man  whose  salary  is  $20  a  week  spends  $14^  a  week.    In 
how  many  weeks  can  he  save  $300? 

3.  A  school  paid  $103  for  desks  and  one  chair.    .The  chair 
cost  $4,  and  the  desks  $3  each.     How  many  desks  were  pur- 
chased ? 

4.  A  train  has  just  passed  the  second  station  of  its  trip  of 
95  miles.     The  first  station  is  25  miles  from  the  start;  and  the 
second  station  is  17  miles  from  the  first.     How  many  miles  of 
the  trip  are  left? 

5.  If  a  retailer  pays  $2.75  per  box  of  120  oranges  and  sells 
the  oranges  at  35  cents  a  dozen,  what  does  he  make  on  a  box? 

6.  If  the  retailer  buys  larger  oranges,  96  to  a  box,  at  the  same 
price  and  sells  them  at  50  cents  a  dozen,  how  much  does  he 
make  per  box? 

7.  A  man  gave  away  $2,200  as  follows :    */2  to  his  son,  %  to 
his  daughter,  ^  to  his  brother,  and  the  remainder  to  a  friend. 
How  much  did  the  friend  receive? 

8.  A  dealer  paid  $30  for  a  dozen  pairs  of  shoes ;  he  sold  them 
at  a  gain  of  $.75  a  pair.     What  would  you  have  to  pay  for  a 
pair? 

9.  Bought  a  pony  and  cart  for  $200.    Sold  the  pony  for  $150, 
gaining  $25.     What  did  the  cart  cost  ? 

10.  In  buying  48  cans  of  tomatoes  how  much  is  gained  by 
buying  two  cases  of  2  dozens  each  at  $2.90  a  case,  over  buying 
at  the  rate  of  3  cans  for  45  cents  ? 

11.  A  man  spent  2/$  of  his  money  and  had  $18  left.     How 
much  had  he  at  first? 

12.  The  two  largest  states  in  the  union  are  Texas  and  Cali- 
fornia.    Texas  exceeds   California   in  area  by    106,000   square 
miles.     The  sum  of  their  areas  is  418,000  square  miles.     Find 
the  area  of  each. 


1  This  test  is  not  available  in  printed  form.    Those  desiring  to  do  so  are 
at  liberty  to  print  it. 


SECTION  II 

CONDITIONS  AND  DIRECTIONS  FOR  GIVING  THE 

TESTS 

To  get  the  greatest  benefit  from  using  a  test  the  conditions 
under  which  it  is  given  should  duplicate  those  under  which  it 
was  standardized.  Reasoning  Test  I  is  standardized  in  that 
it  was  given  by  the  author  to  over  three  thousand  pupils  in  one 
hundred  and  fifty-two  classes  of  twenty-six  representative  school 
systems  of  the  United  States.  These  measurements  were  made 
in  1907  and  1908,  and  the  test  is  further  standardized  in  that 
it  has  been  subsequently  given  in  a  number  of  representative 
school  systems.  Test  II  was  developed  by  giving  to  about  five 
hundred  pupils  the  problems  of  Test  I  interspersed  with  other 
problems. 

In  the  schools  tested  by  the  author,  and  presumably  in  the 
subsequent  tests  by  others,  the  conditions  were  under  such  con- 
trol that  they  were  similar  in  each  room  of  each  school. 

POINTS  TO  BE   OBSERVED 

1.  No   announcement   that   a   test   is   to   be   given   should  be 
made  to  pupils. 

2.  All  directions  to  pupils  should  be  given  by  the  tester.     (If 
the  tester  is  a  person  from  outside  the  system  and  if  the  teacher 
and  principal  are  busy  filling  blanks  with  helpful  data,  the  original 
conditions  will  be  more  fully  duplicated.) 

3.  Principals   and    superintendents    should    refrain    not    only 
from  communicating  with  pupils  about  the  tests,  but  also  from 
being  present  in  the  room  either  immediately  before  or  during 
the  test. 

4.  The   time   limit   should   be   kept   exactly — fifteen   minutes 
to  the  second. 

5.  No  mention  should  be  made  to  the  pupils  of  time  limit. 

6.  Each  pupil  should  be  furnished  with  a  copy  of  the  test  and 
should  have  the  printed  side  turned  down  until  all  begin. 

7.  Nothing  should  be  said  to  pupils  about  the  use  of  scratch 
paper,  working  in  steps,  amount  of  work  to  put  down,  etc. 

6 


Conditions  and  Directions  for  Giving  the  Tests  7 

HOW  TO  GIVE  THE  TESTS 

1.  See  to  it  that  the  conditions  are  a  duplication  of  those 
stated  above  in  so  far  as  practicable. 

2.  Use  exactly  the  following  directions  to  pupils. 

1.  Take  the  materials  that  you  usually  take  for  your  arith- 
metic work.    Prepare  two  sheets  of  paper — headings  and  all. 
Have  two  sheets  ready  in  case  you  may  need  them.     Use  pencils. 
Keep  slip  with  printing  turned  down  until  we  are  ready  to  begin. 

2.  Now,  do  you  have  everything  ready?    In  order  for  you 
to  do  your  best  in  this  test,  you  will  need  to  do  just  as  all  the 
other  boys  and  girls  who  have  taken  this  test  have  done.     So 
pay  close  attention  and  do  just  as  I  ask  you  to  do. 

3.  You  will  not  need  to  mark  these  (slips)  papers  at  all 
You  will  find  directions  at  the  top  of  these  (slips)  papers,  telling 
you  just  what  to  do,  so  you  will  not  need  to  ask  any  questions 
and — I  do  not  think  I  need  to  say  this  to  you,  but  I  will,  just 
because  I  have  to  all  the  other  boys  and  girls, — be  especially 
careful  not  to  see  anybody  else's  work.     It  is  not  easy  not  to  see, 
but  if  you  pay  close  attention  to  your  own  work  only,  the  test 
will  be  the  best. 

4.  Begin.     (Allow  exactly  fifteen  minutes.) 

In  order  that  these  directions  may  be  followed  verbatim  they 
have  been  printed  on  cards.  These  cards  may  be  secured  from 
the  Bureau  of  Publications,  Teachers  College. 


SECTION  III 
DIRECTIONS  FOR  SCORING  THE  TESTS 

There  are  several  bases  on  which  solutions  might  be  scored. 
Four  that  have  been  used  are  (i)  correct  answers,  (2)  reason- 
ing— not  counting  partial  solutions,  (3)  reasoning — counting 
partial  solutions,  (4)  accuracy  of  reasoning.  Choice  among 
these  should  be  made  according  to  the  purpose  for  which  the 
scores  are  to  be  used.  Scoring  for  answers  is  much  the  most 
expeditious ;  scoring  for  reasoning  increases  the  accuracy  of 
measuring  reasoning  ability  as  such;  and  counting  partial  solu- 
tions increases  the  exactness  of  measuring  the  ability  of  indi- 
viduals. "  Correct  answers  "  is  the  best  basis  for  survey  pur- 
poses ;  "  reasoning — not  counting  partial  solutions,"  is  best  for 
class  diagnosis  and  general  supervision  purposes ;  and  "  reason- 
ing— counting  partial  solutions  "  is  much  the  best  for  individual 
pupil  diagnosis  and  teaching  purposes.  Accuracy  in  work  is 
quite  as  important  as  amount  of  work.  Hence,  accuracy  is  essen- 
tial for  all  purposes.  Standards  for  interpreting  scores  com- 
puted on  each  of  these  bases  will  be  found  in  Section  IV. 

In  order  to  have  the  scores  of  a  system,  school,  class,  or 
pupil  comparable  with  the  author's  standards,  it  is  essential  that 
the  solutions  be  scored  in  the  same  way  as  were  those  by  which 
standards  were  set.  To  this  end,  it  is  necessary  that  the  follow- 
ing directions  be  followed  precisely. 

I.      SCORING  ON   BASIS  OF  CORRECT  ANSWERS 

When  "  correct  answers  "  is  the  basis  for  scoring,  each  solu- 
tion is  checked  for  correctness  of  answer;  and  correct  answers 
to  the  respective  problems  are  weighted  as  follows : 

Problem  Score          Problem  Score 


1.  ..  . 

.     1 

7         .    . 

.    .    .     1.2 

2  

1 

8     ... 

1  6 

3  

.1 

9 

2 

4  

1 

10 

.   2 

5  

1 

11  

..2 

6.. 

..1.4 

12.. 

..2 

Directions  for  Scoring  the  Tests  9 

For  example,  any  one  of  the  first  five  problems  solved  correctly 
should  score  I  for  the  pupil,  school,  and  system ;  the  sixth  would 
count  1.4;  the  seventh,  1.2,  etc. 

2.       SCORING    ON    BASIS    OF    REASONING NOT    COUNTING    PARTIAL 

SOLUTIONS 

When  reasoning  is  the  basis  for  scoring,  each  solution  is 
checked  for  correctness  of  reasoning,  regardless  of  mistakes  that 
may  have  been  made  in  operations,  handling  fractions,  copying, 
etc.  For  example,  in  the  following  solution  of  problem  5,  Test 
I,  the  mistakes  are  in  addition  and  multiplication,  and  on  the 
basis  of  correct  reasoning  the  solution  was  scored  correct. 

$2.50  $5.50 

2.00  9 


$5.50  cost  of  each  $50.50  answer 

This  pupil  should  have  checks  placed  in  his  diagnostic  blank 
(See  Form  II,  Section  VI),  showing  an  error  in  addition  and  one 
in  multiplication  for  problem  5. 

In  the  following  solution  of  problem   I,  Test  I,  the  mistake 
is  in  copying  or  reading  and  the  solution  was  scored  correct. 


lOOff  —  79£  =  2l£  answer 

In  this  case  the  pupil's  diagnostic  blank  should  show  a  check  under 
copying  and  reading  with  a  note  stating  difficulty  of  determining 
whether  the  mistake  was  in  copying  or  in  reading. 

The  weightings  for  correct  reasoning  of  the  respective  prob- 
lems are  the  same  as  those  listed  above  for  correct  answers. 

On  this  basis,  no  credit  is  given  for  partial  solutions.  This 
procedure  has  been  found  sufficiently  accurate  for  group  meas- 
urements when  the  median  is  used  as  the  measure  of  the  scores 
of  the  group,  or  when  the  scores  are  tabulated  in  the  form  of 
the  per  cents  of  pupils  making  respective  scores.  (See  Figs. 
2,  3,  and  5,  Section  V.) 


io  Standardised  Reasoning  Tests 

3-      SCORING   ON    BASIS   OF   REASONING COUNTING    PARTIAL 

SOLUTIONS 

When  partial  solutions  are  counted  the  procedure  in  scoring 
is  similar  to  that  above  except  that  when  part  of  a  solution  is 
wrong  or  incomplete,  credit  is  given  for  the  steps  that  are 
reasoned  correctly.  For  example,  one  step  in  the  solution  of 
problem  i,  Test  I,  shown  below  is  correct. 

7i          65i 
X2        —  U{ 

14i  41£ 

Here  l/z  credit  was  allowed. 

If  a  problem  is  unfinished,  credit  is  given  for  the  steps  taken 
if  they  are  correct,  or  in  so  far  as  they  are  correct.  Two  partially 
completed  solutions  will  illustrate : 

(a)  Partial  solution  of  problem  6,  Test  I : 

2200  pupils. 

1/2  of  2200  is  1100  pupils  in  primary  grades. 

This  was  counted  1/5  correct  and  the  problem  was  scored  1/5 
of  1.4  or  .28. 

(b)  Partial  solution  of  problem  5,  Test  I : 

$2.50  cost  of  one  uniform. 
9 


$22.50  cost  of  nine  uniforms. 

$2.00  cost  of  one  pair  of  shoes. 
9 


$18.00  cost  of  nine  pairs  of  shoes. 

This  was  counted  2/$  correct  and  the  problem  was  scored  2/$  of 
i  or  .67.  On  this  basis,  credit  is  given  for  all  partial  solutions. 
These  procedures  are  essential  for  the  exactness  needed  in  secur- 
ing measures  of  the  reasoning  abilities  of  the  individual  pupils. 

4.       SCORING    FOR    ACCURACY 

Accuracy  is  scored  only  on  the  basis  of  reasoning.  For  fifty 
or  more  pupils,  partial  solutions  do  not  need  to  be  counted ;  but 
for  less  than  fifty  pupils  the  increase  in  exactness  justifies  count- 
ing partial  solutions. 

Accuracy  is  computed  in  per  cent  and  problems  are  not 
weighted  for  this  purpose.  That  is,  per  cent  of  accuracy  is  found 


Directions  for  Scoring  the  Tests  n 

by  dividing  the  number  of  problems  reasoned  correctly  by  the 
number  attempted.  For  example,  a  group  that  reasoned  330 
problems  correctly  and  attempted  405  problems  would  have  an 
accuracy  score  of  81.5  per  cent;  and  an  individual  pupil  that 
reasoned  5^  problems  correctly  and  attempted  8  problems  would 
have  an  accuracy  score  of  65.6  per  cent. 

AIDS  TO   SCORING  TESTS   I  AND   II 

It  is  usually  better  to  go  through  a  set  of  papers  marking  one 
problem  at  a  time. 

If  an  answer  is  wrong  it  is  often  difficult  to  decide  how  much 
credit  to  allow.  This  is  especially  true  where  the  solution  is 
only  partially  correct,  or  when  a  cumbersome  method  has  been 
employed.  An  essential  precaution  is  to  score  each  problem  or 
step  according  to  the  method  by  which  it  is  worked.  Credit  is 
to  be  given  for  each  correct  step  regardless  of  how  cumbersome 
the  method  may  be.  (The  use  of  a  cumbersome  method  should 
be  noted  for  each  pupil  on  his  diagnosis  blank  and  he  should  be 
taught  to  work  by  more  effective  methods.) 

While  getting  accustomed  to  scoring  it  will  be  found  helpful 
to  refer  to  the  following  indication  of  steps,  especially  for  the 
more  intricate  problems. 

For  guidance  in  scoring  solutions  that  contain  mistakes  in 
operations,  copying,  etc.,  see  the  foregoing  illustrations. 


Scores  for 

AIDS  TO  SCORING  TEST  I  FUH 

Credit 

1.  If  you  buy  2  tablets  at  7  cents  each  and  a  book  for  65  cents, 
how  much  change  should  you  receive  from  a  two-dollar  bill? 

3  step  answers:   U£,   79£,   $1.21  1 

2.  John  sold  4  Saturday  Evening  Posts  at  5  cents  each.     He  kept 
1/2  the  money,  and  with  the  other  1/2  he  bought  Sunday  papers  at 
2  cents  each.    How  many  did  he  buy? 

3  step  answers:  20£,  lO^f,  5  papers  1 

3.  If  James  had  4  times  as  much  money  as  George,  he  would  have 
$16.    How  much  money  has  George? 

1  step  answer:  $4  1 

4.  How  many  pencils  can  you  buy  for  50  cents  at  the  rate  of  2  for 
5  cents? 

2  step  answers:  10  (times),  20  pencils  1 
Or  2  step  answers:  2%£,  20  (times) 

5.  The  uniforms  for  a  baseball  nine  cost  $2.50  each.    The  shoes 
cost  $2  a  pair.    What  was  the  total  cost  of  the  uniforms  and  shoes 
for  the  nine? 

3  step  answers:    $22.50,   $18.00,   $40.50  1 
Or  2  step  answers:    $4.50,  $40.50 

6.  In  the  schools  of  a  certain  city  there  are  2,200  pupils;  1/2  are  in 
the  primary  grades,  1/4  in  the  grammar  grades,  1/8  in  the  high  school, 
and  the  rest  in  the  night  school.    How  many  pupils  are  there  in  the 
night  school? 

5  step  answers:  1,100,  550,  275,  1925,  275  pupils  1.4 

Or  5  step  answers:  1/2,  1/4,  1/8,  leaves  1/8  =  275  pupils 

7.  If  3|  tons  of  coal  cost  $21,  what  will  5£  tons  cost? 

2  step  answers:  $6.00,  $33.00  1.2 

8.  A  newsdealer  bought  some  magazines  for  $1.    He  sold  them  for 
$1.20,  gaining  5  cents  on  each  magazine.    How  many  magazines  were 
there? 

2  step  answers:  20jf,  4  magazines  1.6 

9.  A  girl  spent  1/8  of  her  money  for  carfare,  and  three  times  as 
much  for  clothes.    Half  of  what  she  had  left  was  80  cents.    How 
much  money  did  she  have  at  first? 

4  step  answers:  3/8  clothes,  4/8  carfare  and  clothes,  $1.60, 

$3.20  2 

10.  Two  girls  receive  $2.10  for  making  button-holes.    One  makes 
42,  the  other  28.    How  shall  they  divide  the  money? 

4  step  answers:  70,  $.03,  $1.26,   $.84  2 

42        6        3    28       2 

Or  5  step  answers:  70,     —  =  —  =  — ,  —  =  — , 
70      10       5    70       5 
3  2 

—  of  $2.10  =$1.26,    —  =  84^ 
5  5 

11.  Mr.  Brown  paid  1/3  of  the  cost  of  a  building;  Mr.  Johnson 
paid  1/2  the  cost.    Mr.  Johnson  received  $500  more  annual  rent  than 
Mr.  Brown.    How  much  did  each  receive? 

4  step  answers:   1/2—1/3=1/6,  $3000  rent, 

$1000  =  B's,  $1500  =  J's  2 

12.  A  freight  train  left  Albany  for  New  York  at  6  o'clock.    An 
express  left  on  the  same  track  at  8  o'clock.     It  went  at  the  rate  of  40 
miles  an  hour.    At  what  time  of  day  will  it  overtake  the  freight  train 
if  the  freight  stops  after  it  has  gone  56  miles? 

3  step  answers:  1 2/5  hr.,  1 2/5  hr.  =  1  hr.  24  min.,  8  o'clock 

plus  1  hr.  24  min.  =  9.24  2 

12 


Scores  for 

AIDS  TO  SCORING  TEST  II  F«B 

Credit 

1.  A  man  starting  on  a  journey  took  $200.    He  paid  for  railroad 
fare  $67;  for  berth  in  sleeping  car,  4  days,  $2  a  day;  hotel  bills,  15  days, 
$3  a  day  ;  other  expenses,  $25.    How  much  money  had  he  left? 

4  step  answers:  $8,  $45,  $145,  $55  1 

2.  Sam  had  12  marbles.    He  found  3  more  and  then  gave  6  to 
George.    How  many  did  Sam  have  left? 

2  step  answers:  15  marbles,  9  marbles  1 
Or  2  step  answers:  3  marbles,  9  marbles 

3.  A  man  bought  163  barrels  of  flour  at  $11  a  barrel.    Fifteen 
barrels  were  spoiled  and  the  remainder  sold  at  $13  a  barrel.    Did  he 
gain  or  lose  and  how  much? 

4  step  answers:  $1793,  148  bbl.,  $1924,  $131  gain 
Or  5  step  answers:  $165  loss  on  15  bbl.,  148  bbl.,  $296,  $165, 
$131  gain  1 

4.  A  drover  bought  132  head  of  cattle  at  $45  a  head  and  67  at  $61  a 
head.    He  sold  them  at  $50  a  head.    Did  he  gain  or  lose  and  how 

much?         6  step  answers:  $5940,   $4087,   $10,027,  199  head,  $9950, 

$77  loss  1 

Or  5  step  answers:  $5,  $660,  $11,  $737,  $77  loss 

5.  A  boy  had  $500  when  he  was  16  years  old,  and  he  saved  $100 
a  year  until  he  was  20.    How  much  had  he  then? 

3  step  answers:  4  yrs.,  $400,  $900  1 

6.  If  a  bookkeeper  receives  $1400  a  year  for  his  services  and  his 
expenses  are  $840,  in  what  time  can  he  save  enough  to  buy  42  acres 
of  land  at  $140  an  acre? 

3  step  answers:  $560,  $5880,  10  £  years  1.4 

Or  3  step  answers:  $560,  4  acres  per  year,  10  j  years 

7.  Mr.  Brown  earns  $32  while  Mr.  Smith  earns  $21.     How  much 
money  will  Mr.  Brown  earn  while  Mr.  Smith  earns  $189? 

2  step  answers:  9  periods  of  time,  $288  1  .  2 

Or  2  step  answers:  $189=  9  X  $21,  9  X  $32=  $288 

32    32 
Or  2  step  answers:  B.  earns  —  ,  —  X  189  =  $288 

21    21 

8.  A  school  in  a  certain  city  used  2516  pieces  of  chalk  in  37  days. 
Three  new  rooms  were  opened,  each  holding  50  children,  and  the 
school  was  then  found  to  use  87  pieces  of  chalk  per  day.    How  many 
more  sticks  of  chalk  were  used  per  day  than  before? 

2  step  answers:  68  pieces,  19  pieces  1  .  6 

9.  Divide  432  between  A.  and  B.  so  that  A.  will  have  $16  more 

111311  B"       3  step  answers:  $416,  B  $208,  A  $224 

Or  4  step  answers:  $216,  $8,  B  $208,  A  $224  2 

10.  Two  carpenters  receive  $154  for  the  work  they  do  on  a  house. 
One  worked  18  days  and  the  other  26  days.    How  much  did  each 
receive?      ^  stgp  answers:  44  d^'  work>  $3^  $91>  §63  2 


11.  Mr.  Smith  owns  a  1/4  interest  in  a  taxi  line  and  Mr.  Jones 
owns  the  rest.    Mr.  Jones  receives  $980  more  a  year  than  does  Mr. 
Smith.    How  much  does  each  receive? 

4  step  answers:  3/4,  1/2,  $490  Mr.  Smith,  $1470  Mr.  Jones 

12.  A  man  leaves  a  certain  place  at  8:00  A.  M.  and  travels  due 
east  at  the  rate  of  20  miles  per  hour.    One  hour  later  another  man 
follows  in  the  same  direction  at  the  rate  of  30  miles  per  hour.    What 
time  will  it  be  when  the  second  man  catches  the  first? 

3  step  answers:  2  hr.,  3  hr.,  11:00  A.  M. 

13 


SECTION  IV 
STANDARDS 

Standards  are  needed  for  each  of  the  bases  on  which  solutions 
are  scored.  The  two  sources  of  data  for  setting  these  standards 
are  (i)  scores  from  large  numbers  of  pupils,  and  (2)  results 
of  diagnostic  measuring  and  remedial  teaching.  In  all,  scores 
have  been  received  from  over  100,000  pupils  in  approximately 
500  schools  of  100  cities ;  and  the  diagnostic  measuring  and 
remedial  teaching  were  conducted  under  the  supervision  of  the 
author. 

First  measurements  are  not  the  best  evidence  by  which  to 
judge  a  system,  school,  teacher  or  pupil.  Safer  and  more  helpful 
conclusions  will  be  reached  from  measurements  made  after 
remedial  teaching.  Hence  each  of  the  following  tables  shows 
standards  for  both  before  and  after  remedial  teaching.  The  after- 
remedial-teaching  standards  are  set  for  the  completion  of  the 
respective  grades ;  and  teachers  should  ordinarily  be  allowed  at 
least  three  months  for  improvement  before  judgment  is  pro- 
nounced as  to  results. 

TABLE  I 

STANDARD  MEDIAN  SCORES  on  BASIS  of  ANSWERS 

GRADE 
V      VI    VII  VIII 

Before  diagnostic  testing  and  remedial  teaching 4.0    5.0    6.5    7. 75 

After  diagnostic  testing  and  remedial  teaching 5.0    6.0    7.0    8. 25 

Solutions  will  ordinarily  be  scored  for  answers  only,  for  survey 
purposes,  or  when  a  general  measure  of  arithmetical  ability  is 
wanted.  In  this  case  the  median1  is  the  best  measure.  Hence  the 
above  standards  are  based  on  the  average  of  median  scores  from 

1  The  median  is  the  middle  score.  Unless  the  number  of  pupils  is  more 
than  loo,  it  is  usually  simplest  to  find  the  median  by  counting  to  the  middle 
score.  Arrange  the  papers  in  order  of  scores  and  count  off  half  of  the 
scores.  If  there  is  an  odd  number  of  pupils  the  median  score  is  readily 
reached.  If  there  is  an  even  number  of  pupils,  the  median  is  the  average 
of  the  two  middle  scores. 

14 


Standards  i 5 

a  large  number  of  schools.    The  average  of  the  medians  from  77 
systems  (approximately  400  schools)  were: 

Grade  V  4.0,  VI  4.9,  VII  6.4,  VIII  7.7 
Some  of  the  cities  that  made  the  better  scores  are: 

GRADE 

City                                                              Date    V     VI      VII  VIII 

Lead,S.D .1915    4.5    6.1    9.2  11.4 

Salt  Lake,  Utah 1915    4.3    6.9    9.1  11.0 

Cold  Springs,  N.  Y 1917     ...     6.1    9.5  11.3 

The  two  cities  that  made  the  lowest  scores  were: 

GRADE 
V     VI  VII  VIII 

AFarWestCity 3.0    2.9    4.3    5.6 

A  Southern  City 2.5    4.4    5.6    4.9 

TABLE   II 
STANDARD  SCORES  on  BASIS  of  REASONING 

GRADE 

V       VI      VII       VIII 

Before  diagnostic  testing  and  remedial  teaching ..  4.75    5.75    7.0        8.0 
After  diagnostic  testing  and  remedial  teaching  ...5.5      6.5      7 . 75      8 . 75 

The  standards  of  Table  II  will  suffice  both  when  the  reasoning 
of  partial  solutions  is  counted,  and  when  only  completed  solu- 
tions are  counted.  When  only  completed  solutions  are  counted, 
the  measurement  is  ordinarily  for  the  purpose  of  group  com- 
parison and  the  median  is  the  best  measure. 

These  standards  were  based  on  the  average  of  median  scores 
from  some  100  schools  in  35  cities.  These  averages  were: 
Grade  V  4.7,  VI  5.8,  VII  7.0,  VIII  8.0. 

The  city  with  the  highest  scores  was  Council  Bluffs  (1918), 
with  the  following  scores :  Grade  V  4.1, VI  6.2,  VII  7.3,  VIII  10.3. 

TABLE  III 
STANDARDS  FOR  ACCURACY  OF  REASONING  (TENTATIVE) 

GRADE 

V        VI      VII      VIII 

Before  diagnostic  testing  and  remedial  teaching. .  70%    75%    80%    85% 
After  diagnostic  testing  and  remedial  teaching ....  75        80        85       90 

Accuracy  in  work  is  quite  as  important  as  the  amount  of 
work.  Hence  standards  are  needed  for  accuracy  as  well  as  for 


16  Standardised  Reasoning  Tests 

scores.  As  yet,  comparatively  little  attention  has  been  given 
to  this  phase  of  measuring.  The  above  scores  are  therefore 
tentative.  They  are  based  on  the  author's  measurement  of  the 
accuracy  of  Grade  VI  pupils  in  some  thirty  representative  school 
systems.  The  accuracy  of  reasoning  among  these  systems  was 
found  to  range  from  54.9%  to  85.6%,  the  median  being  72.1%. 
The  author  also  followed  the  remedial  teaching  of  a  typical 
Grade  VI  class  for  a  term.  During  that  time  the  accuracy  was 
improved  from  a  median  of  74.5%  to  a  median  of  84.5%. 


SECTION  V 
REPRESENTING  SCORES 

The  degree  of  benefit  derived  from  giving  the  tests  will  turn 
on  the  effectiveness  with  which  the  results  are  utilized.  One 
of  the  essentials  for  full  utilization  is  effective  representation. 
There  are  many  effective  means  of  representing  scores  and  the 
one  which  a  given  person  uses  should  be  the  one  that  is  most 
effective  for  that  person.  For  some,  this  will  be  tables  of  scores ; 
for  others,  surfaces  of  distribution;  for  others,  graphs  of  prog- 
ress. Simple  graphs  are  not  difficult  to  construct  and  a  little 
practice  will  help  most  persons  to  use  them  to  great  advantage. 
Whatever  scheme  is  used  the  result  should  be  that  the  scores 
are  so  placed  that  they  may  be  readily  and  fully  interpreted. 

A  simple  though  not  especially  effective  representation  is  that 
of  placing  the  score  of  a  given  system  with  that  of  other  repre- 
sentative systems.  The  following  is  an  illustration  of  this  repre- 
sentation as  used  in  the  Springfield,  Illinois,  Survey: 

SCORE  PER  EACH  100  PUPILS  IN  REASONING  IN  ARITHMETIC  IN 
SPRINGFIELD  AND  26  OTHER  SCHOOL  SYSTEMS 


Spring- 

Spring- 
field's 

Lowest 

Middle 

Highest 

field 

rank 

from  top 

Reasoning  

356 

550 

914 

508 

19 

Accuracy.  . 

55 

72 

86 

70 

19 

The  figures  in  the  last  column  of  the  table  show  that  in  both  the  amount 
of  work  accomplished  and  the  accuracy  with  which  it  was  done  the  Spring- 
field children  rank  in  the  iQth  place  among  the  27  systems  compared. 
That  is  to  say,  they  are  more  than  two-thirds  of  the  way  down  the  list. 

The  purpose  of  the  representation  is  the  best  criterion  for 
deciding  which  of  the  various  schemes  of  representation  to  em- 
ploy. The  main  purposes  for  which  scores  are  represented  are 
to  aid  supervision  and  to  aid  teaching.  Superintendents  and 
other  supervisors  responsible  for  the  work  of  many  teachers 
need  to  be  able  to  see  readily  the  status  of  a  class  as  a  whole; 
teachers,  on  the  other  hand,  need  to  be  able  to  see  readily  the 
status  of  pupils  as  individuals. 

17 


i8  Standardised  Reasoning  Tests 

REPRESENTING   SCORES  AS   AIDS   IN    CLASS   DIAGNOSIS   AND  GENERAL 

SUPERVISION 

For  purposes  of  supervision  the  ideal  representation  is  that 
which  conveys  the  class  status  with  the  minimum  of  time  and 
effort.  One  of  the  best  plans  is  that  of  the  per  cent  distribution 
according  to  scores.  This  is  illustrated  in  the  portrayal  of  the 
Butte,  Montana,  reasoning  scores  (Fig.  2).  Fig.  2  represents 
the  percentage  of  children  making  the  given  scores  in  reasoning 
problems.  For  example,  19  per  cent  of  the  fifth  grade  children 
made  score  of  o;  19  per  cent  made  score  of  i;  etc.  The  lines 
representing  the  median  scores  for  each  grade  tell  about  how 
many  in  each  grade  surpass  the  median  scores  for  the  grades 
above,  and  how  many  fall  below  the  median  scores  for  the 
grades  below. 

Representation  by  graphs  will  be  facilitated  by  using  co- 
ordinate paper  of  suitable  rulings.  The  following  form  shows 
a  good  size. 


20 

at 

18' 


'6 


Pi 

IZ 

m 

10  L 

8 

<b 

a  g 

e 


Scores 


0       /       2      3      4-     5      fe      T      8      5>      /O     //      li    13    I*    15    16 

FIG.  i.     Graph   form   for  representing  percentages  of  pupils  attaining 
given  scores. 


Representing  Scores 


MEDIAN  SCORES 

6TH       7TH     8TH 
2.2       3.9         -S.9        7.7 


STH  GRADE 


SCORES    0      I      2     3     A     5     6     7     8     9    10    II     12    13    14  15 


FIG.  2.    Representing  percentage  of  children  making  the  given  scores  in 
reasoning  problems.     Butte,  Montana. 


Standardised  Reasoning  Tests 

I 


FIG.  3.  Percentages  of  pupils  making  various  scores.  (75  VI  A  pupils, 
Bloomington,  Ind.) 

The  procedure  in  filling  out  a  form  to  represent  the  percentages 
of  a  given  system  or  class  may  be  explained  in  connection  with 
Figs,  i  and  3.  The  steps  are : 

1.  Sort   the   papers,   placing   the   lowest   score    on   top   and 
graduate  so  that  the  highest  is  on  the  bottom. 

2.  Compute  the  per  cents  of  pupils  that  make  the  respective 
scores  of  o  (o  to  J/£),  i  (l/2  to  1^2),  2  (\y2  to  2^2),  etc. 

3.  Construct  a  surface  of  distribution  showing  the  frequency 
of  per  cents  and  the  deviation  from  the  standard. 

Step  2  is  illustrated  in  the  following  table  which  shows  the  per- 
centage of  pupils  that  made  the  respective  scores  in  the  VI  A 
class  in  Bloomington,  Indiana,  in  1914. 
Number  grading 


r 

0— 

5, 

1  = 

1. 

3%  of  the 

75  pupils. 

5— 

1 

5, 

1  = 

1 

3% 

» 

a 

« 

u 

1 

5— 

2 

5, 

4     

1, 

3% 

(i 

K 

« 

u 

2. 

5— 

3 

.5, 

0  = 

0, 

0% 

« 

U 

a 

u 

3 

.5— 

4 

5, 

e  

6 

.6% 

« 

* 

" 

u 

4 

,5— 

5. 

,5, 

7  = 

9 

.3% 

a 

« 

u 

u 

5 

,5— 

6 

5, 

14  = 

18. 

6% 

u 

« 

u 

u 

6 

,5— 

7 

,5, 

14  = 

18.6% 

u 

II 

u 

u 

7.5—  8.5,     7=    9.3% 


Representing  Scores  21 

Number  grading  8.5—9.5,  4  =    5.3%  of  the  75  pupils 

"  9.5—10.5,  10=13.3%"     "     •        « 

"             "  10.5—11.5,  7=    9.3%  «     "     " 

«  11.5—12.5,  1=    1.3%  •     •     • 

"  12.5—13.5,  3=    4.0%  «     «     «        « 

Fig.  3  illustrates  the  third  step.  By  comparing  the  above  table 
with  the  graph  in  Fig.  3  it  will  be  seen  that  one  pupil,  or  1.3 
per  cent  of  the  75  pupils,  whose  score  was  o  to  .5,  is  represented 
in  the  lower  left-hand  corner  of  the  form;  the  1.3  per  cent  with 
a  score  of  .5 — 1.5,  is  represented  next;  the  1.3  per  cent  with  a 
score  of  1.5 — 2.5  next;  and  as  there  were  no  pupils  receiving  a 
score  of  2.5 — 3.5  there  is  no  representation  on  the  form;  but 
6.6  per  cent  had  a  score  of  4  (3.5 — 4.5)  and  the  graph  extends 
above  4  to  6.6 ;  the  9.3  per  cent  with  a  score  of  5  are  represented 
above  5,  etc.  The  vertical  broken  line  at  6.5  indicates  the  Stand- 
ard. Thus  the  supervisor  can  see  at  a  glance  ( I )  the  percentage 
of  pupils  attaining  each  score,  and  (2)  the  percentages  that  are 
up  to  or  above  the  standard. 

Another  way  of  representing  the  data  for  supervisory  pur- 
poses is  illustrated  in  the  following  graph  of  scores  of  the 
Bloomington,  Indiana,  schools  where  the  author  has  tested  the 
reasoning  abilities  of  pupils  for  a  series  of  years. 


- 


too 


ffoe 

3X6 


I4?  10  1112.  iiu 

FIG.  4.    Improvement  in  reasoning,  Bloomington,  Indiana,  Schools,  1910-1914. 
C.=Central  School     Mc.=McAlla  School      F.=Fairview  School      All=Average  of  three 


22 


Standardised  Reasoning  Tests 


J* 


-SOT 


0       I 


FIG.  5.  Reasoning  abilities,  Iowa  State  Teachers  College  Training 
School,  January,  1916.  This  representation  enabled  the  supervisor  to  see 
at  a  glance  where  the  pupils  of  each  of  these  classes  stood  in  reasoning  at 
the  midyear  and  to  know  what  percentage  of  each  class  needed  special 
attention  during  the  second  semester. 


Representing  Scores  23 

Another  markedly  effective  illustration  is  that  employed  by 
Superintendent  Stark  of  the  Hackensack,  New  Jersey,  schools 
(Fig.  6.)  to  represent  gains  during  the  years  1911  to  1915. 


6 


O 

o 

ui 

K 


UJ      -J 

O    \~ 

\-      2 
CO     ««J 

r 

Q 

Z 
D 
b. 


FIG.  6. 


Standardised  Reasoning  Tests 


REPRESENTING  SCORES  AS  AIDS  IN   INDIVIDUAL  DIAGNOSIS  AND 
REMEDIAL  TEACHING 

For  purposes  of  teaching,  the  ideal  representation  is  that  which 
conveys  the  status  of  individual  pupils  with  the  minimum  of 
time  and  effort.  As  far  as  the  author  has  found,  the  best  form 
of  graph  for  this  purpose  is  that  which  shows  the  exact  status 
of  each  individual  pupil.  As  the  supervisor  needs  to  see  by 
classes,  so  the  teacher  needs  to  see  by  individuals.  The  follow- 
ing are  illustrations. 


7 


«?3 


FIG.  7.  Score  of  individual  pupils.  VI  A-i  Bloomington,  Ind.,  1914.  Each 
line  stands  for  the  score  of  a  pupil.  For  purposes  of  identification  the 
initials  of  the  pupils  are  placed  at  the  bottom,  e.g.,  J.H.  not  only  reached 
the  standard,  but  did  so  well  that  his  line  extends  almost  twice  as  high  as 
the  standard  of  the  VI  A  grade  requires.  This  form  of  graph  brings  out 
vividly  just  where  each  pupil  stands,  e.g.,  E.L.  with  a  score  of  less  than  I. 


Representing  Scores 


7* 


7* 
6* 
f* 

4? 


Af 


JO 


y 


^L= 
*:nc. 


^K 


X  ^oMI(r^l««l  ol  *J 

^o:^  ao  V^P> 


FIG.  8.  Per  cents  of  accuracy  of  individual  pupils.  VI  A-i,  Bloomington 
Ind.  Here  the  lines  stand  for  per  cents  of  accuracy  for  individual  pupils, 
e.g.,  D.C.  is  well  above  the  Standard  in  accuracy,  while  E.L.  is  far  below. 


26 


Standardised  Reasoning  Tests 


FIG.  9.  Pupils  making  various  scores  and  accuracy  per  cents.  VI  A-i 
(21  pupils)  Rloomington,  Ind.  This  is  a  representation  of  both  scores 
and  accuracy  for  individual  pupils.  It  enables  the  teacher  to  see  at  a 
glance  the  status  of  each  pupil  in  both  amount  and  quality  of  work,  e.g., 
J.H.  is  seen  to  be  above  Standard  in  both,  while  L.D.  is  markedly  accurate 
but  does  not  reach  the  Standard  in  score.  This  shows  the  teacher  that 
L.D.'s  difficulty  is  lack  of  speed,  and  she  is  in  a  position  to  determine  the 
cause  of  the  slowness  and  to  set  about  removing  it. 


SECTION  VI 
UTILIZING  THE  RESULTS 

Educational  measurements  are  ordinarily  made  for  one  or 
more  of  three  purposes,  viz.,  survey,  supervision,  or  teaching. 
In  surveys  the  purpose  is  served  when  the  measurement  is 
accurately  made,  effectively  portrayed,  and  adequately  inter- 
preted. In  supervision  the  purpose  is  partially  served  by  these 
same  steps,  but  if  supervision  is  to  render  its  best  service,  the 
results  must  be  further  utilized.  They  must  be  utilized  for 
improving  teaching.  And  in  teaching  the  dominant  purpose 
of  measurements  should  always  be  improvement. 

How,  then,  may  the  results  of  measuring  reasoning  ability 
in  arithmetic  be  utilized  for  the  improvement  of  teaching  reason- 
ing in  that  subject?  The  best  answer  to  this  question  is  found 
in  the  parallel  between  the  work  of  the  good  teacher  and  the 
work  of  the  good  physician,  or  good  hygiene  worker.  The  good 
physician  measures,  diagnoses,  prescribes  treatment,  and  meas- 
ures again.  If  the  case  is  a  serious  one  he  secures  the  help 
of  a  competent  nurse.  So,  the  good  teacher  with  the  help  of 
the  supervisor,  will  measure,  diagnose,  treat,  and  measure  again. 

As  a  means  of  facilitating  the  work  of  remedial  teaching,  two 
forms  of  diagnostic  record  sheets  are  offered.  Form  I  is  a  class 
diagnostic  sheet.  It  is  for  the  purpose  of  locating  class  tendencies 
and  needs.  Form  II  is  an  individual  diagnostic  blank.  It  is  for 
the  purpose  of  locating  individual  pupil  tendencies  and  needs. 
These  two  forms  are  shown  on  pages  29  and  30. 

Each  of  the  diagnostic  forms  has  been  filled  by  entering  the 
record  of  pupil  G.S.  He  is  represented  as  pupil  I  on  Form  I 
record  sheet.  This  entry  shows  that  G.S.  reasoned  the  first 
six  problems  correctly  but  made  some  mechanical  error  that  gave 
a  wrong  answer  to  problem  i.  This  record  shows  further  that 
G.S.  failed  in  the  reasoning  of  problems  7  to  n,  and  did  not 
attempt  problem  12. 

27 


28  Standardised  Reasoning  Tests 

SAMPLE  SOLUTIONS  OF  REASONING  PROBLEMS—  TEST  I 


Name  of  Pupil  ~3.  of.  Date    c&x£  a.  i.  ,  <?  a  /  _ 

Hame  of  Teacher    T>W<1.  .  Q.       '  Grade        d>  &          Age  13-10 


FTT4  Ol  "cents  back.  10  / 

_66  j> 

¥779         CK  =   C.  20  pencllB 


2.        B^ 

5.       $2.50 


1760 

I  1  of  2fi,,$.10 9 

I  2         i  $40.50  The  coat 


Papa  re 

275 


Zf  I  3.10 

6.        .,.--„,„„ 

/.  H~ 

I        3.       1  of  ifr3  $4  2200 

'  -4-         1 

in  nl£ht  school 


7.        34  .  $21    1  of  Sl_  21  X-JZ.  -  iil  »  73 
1    "         ~  2          2"         2  2 

10. 


of  J2.10  a   $1.  55  each 


i  nso 

24  magazines 


11. 

1  of   $froe&  =   §166. 66-1-     each,    f) 
9.        60  *-* 


01  of  51.66=  20   she  opent   for  car  fare. 


& 

$.20 


60  for  oloathee. 


/L£-ci^a-^>-v(^»i-< 

.    ^.57.. 
/ 


Utilizing  the  Results 


FORM  I.  DIAGNOSTIC  CLASS  RECORD  SHEET 

Date  A/*  ifrl 
CityCtu££o^i^x>v  State  LAfaak,.  School  >Hax^u  Qrade3Z  Claee  &>  Teet  Ho.  -± 
Claaa  Teaoher"VnAxz,  C  Scored  ^ylutMf^^^tUa^rn^  (Score-Correct  in  reaeonlng  and  anawer. 
L  .)Score  wlt]v/=  Correct  reaeonlng  but  wrong  anewer. 
Claee  Median  Score  Olaea  Median  Accuracy  g  )0=wrong  in  both  reaaoning  and  anawer. 
U'Not  attempted. 

o       § 

•  MB 

ft    '5 

</}          CO 
K        3 

ft 
j=     S 

0  0  W 

fepoJS 

M 

<J     .S 

«  §  • 

eS 

=B  2  c 

c    ."" 

.       3   ft  ?>> 

"•  v-nj  ivaownvu  IIL.\.I  i_ui  i  tv.nj'  duswcicu  —  niucA  ui  class  accuracy  in  co 

of  the  respective  problems. 

Ho. 

attempted 

o 

Totale 

1  Record  each  pupil's  age  below  his  number.  Record  as  10  until  11,  n 
Per  cent  accuracy  =  Per  cent  accuracy  in  reasoning.  See  Section  III, 
3  Number  attempted  —  Number  correctly  reasoned  =  Index  of  class  accurac 
tive  problems. 

MlimKpr  r>nrra>>t1ir  roo  c/inorl  \Tnrv<Ka.-  ^n*-~t>^ti,r  I  T_J  _r  _1  

A 
IO 

00 

r- 

(0 

8 

n 

IQ 

S 

to 

IO 

CM 
IO 

H 

IO 

O 

n 

—  ^ 

rl 

• 

_.  —  —  - 

•  —  - 
-  —  | 

—  ^ 

-  —  1 

•—•  -. 

•  —  , 
•*-~_ 

— 

•  —  • 

—  - 
—  ^ 

—  . 

— 
.  —  -i 

—  - 
.  —  - 

^  ' 

_-.  — 

—  —  —  ' 

.-  

,  • 

--^ 

^^_ 

O 
H 

Ok 

o 

e> 

o 

» 

* 

IO 

01 

; 

- 

- 

- 

- 

- 

* 

X 

X. 

X 

•^  •> 

c  -^ 

i 

* 

* 

1 

H 

CO 

r 

0 

0 

0 

o  <: 

>   X 

f 
-a 

- 

s9 

to 

B    • 

fs 

Sf> 

~B 

•H 

OJ 

rO 

•* 

iO 

-0 

r- 

0) 

a 

0  r 

H  r- 

i  N 

S 

0 

o 

co 

TJ 

•*» 

04* 

^  C 
0  O 

S-. 

s§ 

*0£  £<±  euiato.ojc  <j 

30 


Standardized  Reasoning  Tests 


The  class  diagnosis  and  treatment  would  depend  on  what  the 
record  shows  for  the  remaining  pupils.  G.S.  evidently  needs 
to  give  more  attention  to  accuracy  in  reasoning.  Should  the 
completed  class  record  show  a  like  tendency  to  try  many  more 
problems  than  are  solved  correctly,  one  prescription  would  be 
more  care  in  checking  answers. 

FORM  II.    INDIVIDUAL  DIAGNOSTIC  RECORD  SHEET 

Scores       Accuracy*     Dates      Test  Ho. 

Grade:2i:a  64  5457.       */a,/a,  T 

School  ^^lu-i-vt^  &.JU- 

Some  of  Pupil  d  e! 


I  i      f  U]J  1 A     f     JO 

0.4. „„,.„..,»«,   „_  v0.<.(5th,   5.5  with  755?  accuracy)  See  aanual  for 

of  rn^nC  )6th     6-5     "       60^         "         (  Stono  R«a8on.   Tests 

of  reasoning  <*.,.,_     n  ne    •       v^.-t         n         f  n\ ...  TW 


(End  of  yaar). 


!7th.   7.75 
3th,   6.75 


, 
Tables    II  and  III 


Ho.  of  Problems.  | 

Score  in  Reasoning 

ITot  ^ttcnptcd 
faulty  Rsaaonlng 
Failure  in  Reading 
~^TnfinlcheA 

uumbersr>s;e  Method 
Undua  Laoel  fr.g 
.JJnneoesaary  Writing 
ton  littlB  Recorded 

Errors       Uaehanioel 
Kn-oro 

ProtaVle 
Cause  of 

(Cocpare 
"P-eason. 
Teats", 

lliBt    Of 

vi.; 

Jraenrl'bed 

Treatment 
Pound 

Helpful. 

Addition 
Suotraotion 

a 

* 
t 

!g 

?"o 

;  -i 

sa 

Handling  Fractions 

I 

-.1 

I 

o 

Borrovrii-j; 

Placing  f&r.  Prod.  6 

Plaair.s  Eaoinp.l  I'oirt 

5 

0 

JL. 

1            J 

y 

&*4*       ®j»  ' 

•VA.4M«Wi/ 

I 

1        T  .) 

» 

5 

I 

J2 

~s~ 

I 

v' 

*?tfi    'hfr^r^t' 

Stlif  Jl**,  I$*Q  A'ftVTf^ 

. 

sg! 

7 

0    _J7_L 

*            ft 

22S 

a 

0        S 

ro_v^2.. 

yX^,'   .^Jj, 

9 

0        -J 

-^- 

i-V    |V      /     )r 

1 
15 

0        / 

"^  i^'-V      C,^ 

0        J 

1*    CT   jT 

X   •/ 

To- 

tails 

1.4        5   1 

I 

' 

1  Per  cent  accuracy  is  a  measure  of  accuracy  in  reasoning  regardless  of 
mistakes  in  operations  or  other  mechanical  errors.  It  is  computed  by 
dividing  the  number  reasoned  correctly  by  the  number  attempted.  Number 
correct  should  be  found  in  Score-in-Reasoning  column  above ;  Number 
attempted,  from  other  columns.  For  computing  accuracy,  see  Section  III, 
p.  10. 

Form  II,  as  filled  for  G.S.,  affords  a  specific  record  of  his 
measurement;  e.g.,  it  indicates  that  his  mistake  in  problem  I 
was  in  copying,  that  he  also  made  a  mistake  in  placing  the 


Utilizing  the  Results  31 

decimal  point  in  problem  4,  although  this  mistake  did  not  affect 
the  answer,  etc. 

Copies  of  these  diagnostic  record  blanks  may  be  secured  from 
the  Bureau  of  Publications,  Teachers  College.  No  device  known 
to  the  author  is  so  helpful  in  the  diagnostic  study  that  is  so 
essential  to  remedial  teaching. 

TWO  DANGERS 

In  interpreting  the  results  of  a  test,  care  should  be  taken  to 
avoid  two  dangers:  (i)  The  danger  of  placing  too  much  reliance 
on  a  single  measure  of  an  individual.  No  pupil  whose  score  is 
surprisingly  low  should  be  so  graded  without  a  second  trial.  The 


FIG.  10.  The  score  of  B.  H.  was  unexpectedly  low  and  to  have  diagnosed 
her  case  on  the  basis  of  the  first  test  would  have  been  inaccurate  and 
unjust. 

first  test  may  have  come  on  an  "  off  day  "  for  the  individual. 
See  Fig.  10.  (2)  The  danger  of  relying  on  the  average  of  the  class 
scores  to  show  whether  the  pupils  are  "  up  to  standard."  It 
may  be  that  there  are  a  few  very  high  scores  and  a  few  very 
low  scores,  and  both  these  undesirable  extremes  will  be  hidden 
in  the  average.  It  is  always  best  to  study  the  scores  of  individual 
pupils,  and,  if  a  single  measure  of  ability  is  needed,  to  use  the 
median. 


32  Standardised  Reasoning  Tests 

CAUSES  OF  FAILURE 

In  diagnosing  cases  of  failure  the  following  are  some  possible 
causes  that  may  well  be  considered: 

1.  Inability  to   read. — Any  pupil  who  is  poor  in  reasoning 
should  be  thoroughly  tested  by  the  use  of  some  of  the 
measures  of  silent  reading. 

2.  Poor  judgment  as  to  the  amount  of  writing  to  do  in  the 
solution. — Much  time  and  energy  are  lost  by  labored  labels 
and  elaborate  indications  of  steps. 

3.  Physical  disability. — Poor  eyesight  is  a  common  hindrance 
in  this  as  well  as  other  phases  of  school  work. 

4.  Lack  of  physical  coordination. — Tests  in  copying  figures 
and  other  coordination  tests  should  be  used. 

5.  Inability  to  see  relationships  between  steps. — This  is  fre- 
quently an  innate  lack  and  very  difficult  to  overcome. 

6.  Mental  laziness. — There  is  no  spur  equal  to  knowing  just 
where  one  stands  and  where  one  ought  to  expect  to  stand. 

7.  Lack  of  a  realization  of  the  passing  of  time. 

8.  Temporary  disability. — Individuals  have  "  off  days  "  and 
surprisingly  low  scores  should  not  be  regarded  as  the  meas- 
ure of  an  individual  without  a  second  trial.1 

SECURING  IMPROVEMENT 

A  record  of  the  treatment  of  certain  individuals  2  with  low 
scores  will  serve  to  illustrate  the  possibility  of  securing  improve- 
ment. The  records  will  be  given  under  diagnosis,  treatment  and 
results. 

Pupil  H.  C. 

Diagnosis:  Up  to  standard  in  reading  ability,  did  not  indulge  in  undue 
labeling,  physical  examination  showed  no  defects,  constantly  made  low 
scores.  Conclusion  as  to  cause  of  low  score:  Mental  laziness  with  lack  of 
realization  of  the  passing  of  time. 

Treatment:  The  pupil  was  first  of  all  made  conscious  of  his  status  by 
comparing  his  score  with  those  of  his  fellow  classmates  and  with  the  stand- 
ard; then  he  was  helped  to  study  his  way  of  working  which  convinced  him 

1  But  as  the  first  scores  are  counted  in  making  the  scale,  it  is  best  that 
all  pupils  be  included  in  the  scores  as  computed  for  the  class  or  system. 

2  This  record  of  pupils  was   furnished  by  Miss  Floe  E.   Correll,  then 
supervising  critic  of  mathematics,  Iowa  State  Teachers  College  Training 
School. 


Utilizing  the  Results 


33 


of  the  seat  of  his  difficulty.  From  day  to  day  lists  of  approximately  equiva- 
lent problems  were  given  him  with  time  limit.  Much  was  made  of  record  of 
scores,  gain  being  expected  by  both  teacher  and  pupil. 

Results:  Within  a  few  days  notable  gain  appeared,  due  to  increased 
ability  to  direct  and  hold  attention  to  the  work  in  hand.  Contrasted  with 
his  previous  tendency  to  wander,  the  pupil  became  capable  of  working  con- 
tinuously in  spite  of  such  distractions  as  people  entering  the  room.  After 
about  twenty  minutes  daily  for  three  weeks  he  raised  his  score  from  4  to 
5.4.  Though  this  is  not  a  large  gain  in  score,  the  boy  had  made  it  largely 
of  his  own  initiative;  he  had  formed  an  ideal  of  concentration,  and  the  con- 
cept of  giving  attention  to  reasoning  processes  was  well  under  way.  It  is 
believed  by  those  who  have  studied  the  boy  that  much  of  his  improvement 
was  due  to  the  convincingness  of  the  objective  evidence  of  his  need  to  improve. 


Some  Pupils  of  a  Certain  Fifth  Grade 

Diagnosis:     Many  pupils  made  very  low  scores,  many  papers  much  cov- 
ered with  such  statements  as,  "  If  one  tablet  cost  7  cents,  2  tablets     ... 
etc."     Here  was  evidently  one  large  source  of  failure. 

Treatment:  Emphasis  was  placed  on  the  possibility  of  saving  time  by  not 
writing  so  much,  brief  labels  were  devised,  originality  was  encouraged,  and 
approval  of  pupils  and  teacher  placed  on  briefest  adequate  statement. 

Results:  As  shown  by  second  test  and  by  daily  work,  much  time  was 
saved  for  reasoning  processes.  The  following  parallel  columns  show  typical 
results. 


Pupil  A.  K. 
In  first  test 
They  would  cost  $18. 

If  one  suit  cost  $2.50,  9  would  cost 
$2.50  X  9  =  $22.50. 
They  would  cost  $40.50. 

Score  in  first  test,  1  1-3 

Pupil  L.  I. 
In  first  test 

If  he  sold  4  papers  and  got  twenty 
cents  for  them,  one-half  would  be  10 
cents  and  with  the  other  10  cents  he 
bought  Sunday  papers,  he  would  buy 
as  many  as  2  will  go  into  10  or 
2J10 

5  papers. 

Score  in  first  test,  3 


In  second  test 
$2.50  X  9  =  $22.50 
$2         X  9  =  $18. 


$40.50. 
Score  in  second  test,  3 


C. 

5 
4 

20 


In  second  test 
One-half  would  be 
10     cents     and     he 
could  buy  five. 


Score  in  second  test,  41-2 


44905 


This  book  is  DUE  on  the  last  date  stamped  below 


Form  L-9-15m-7,'31 


7,08  ANGELES 


QA 
135     Stone  - 

gorf 

Standarized 

reaaoj 

tests  in 
_jarithmetlc 
and)  how  to 


QA 

135 

S87 


UC  SOUTHERN  REGIONAL  LIBRARY  FACILITY 


A    000937152    7 


